Lake Okeechobee System Operating Manual

Lake Okeechobee System Operating Manual
(WQ Subteam)

DRAFT - St Lucie Estuary Nutrient Loading Model

FDEP - Office of Water Policy and Ecosystem Restoration

January 13, 2021

Similar to CRE models
Top down statistical modeling, where we used observed water quality and discharge data to fit a model.
Does not 100% account for C-44 reservior and STA. Could assume a metric ton reduction based on assumed treatment.
Does not account for potential nutrient reductions assoicated with the current Basin Action Management Plan(s).
WQ could extend model period of record to WY2020 (currently CRE and SLE models are set to 2019)

DRAFT
S-80 Water Quality ModelsGoal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
3

S-80 Water Quality Models

Goal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
Period of Record: May 1981 – April 2019 (WY1982 – 2019)

S-80 Water Quality Models

Goal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
Period of Record: May 1981 – April 2019 (WY1982 – 2019)
Parameters of Interest: Total Phosphorus and Total Nitrogen.

S-80 Water Quality Models

Goal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
Period of Record: May 1981 – April 2019 (WY1982 – 2019)
Parameters of Interest: Total Phosphorus and Total Nitrogen.
Predictor Variables: Discharge (S80, S308 and C44 Basin) converted from ft³ s^-1 to Acre-Ft d^-1 and Lake Okeechobee stage elevation were considered.

S-80 Water Quality Models

Goal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
Period of Record: May 1981 – April 2019 (WY1982 – 2019)
Parameters of Interest: Total Phosphorus and Total Nitrogen.
Predictor Variables: Discharge (S80, S308 and C44 Basin) converted from ft³ s^-1 to Acre-Ft d^-1 and Lake Okeechobee stage elevation were considered.
Statistical Modeling:
- Multiple regression models using training and testing datasets (70:30).
  - Training dataset: randomly sampled 70% of monthly data
  - Testing dataset: remaining 30% was used for model testing
- Verified with k-fold cross-validation linear modeling.

S-80 Water Quality Models

Goal: Develop a series of water quality models based on hydrodynamic indicators to be used in planning model scenario evaluation for LOSOM using RSMBN.
Period of Record: May 1981 – April 2019 (WY1982 – 2019)
Parameters of Interest: Total Phosphorus and Total Nitrogen.
Predictor Variables: Discharge (S80, S308 and C44 Basin) converted from ft³ s^-1 to Acre-Ft d^-1 and Lake Okeechobee stage elevation were considered.
Statistical Modeling:
- Multiple regression models using training and testing datasets (70:30).
  - Training dataset: randomly sampled 70% of monthly data
  - Testing dataset: remaining 30% was used for model testing
- Verified with k-fold cross-validation linear modeling.
Consistent with Caloosahatchee River Estuary Nutrient Loading Model.

Set period of record similar to CRE model

C-44 Hydrology

Percent of Basin Discharge
- $Q_{C44}/Q_{S80}$
- Range from 0 to 74.8 %

C-44 Hydrology

Percent of Basin Discharge
- $Q_{C44}/Q_{S80}$
- Range from 0 to 74.8 %

Cumulative discharge (S80) and rainfall (across C44 basin) for the period of May 1979 - Apirl 2019 (WY1980 - 2019) with breakpoints identified using segmented regression.

Rainfall monitoring sites used include S308R, S135R, S80R, PRATT and ACRA2

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q.S80)+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q.S80)+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)
Model assumptions tested and verified (see Model Diagnostics)
- GVLMA (Global Stats = 1.75, $\rho$ =0.78)
Variance inflation factors (VIF) evaluated for model

Variable	VIF
QC44	2.35
QS308	2.15
ln(QS80)	3.53
Mean Lake Stage	2.88

Residuals check for residual autocorrelation (Breusch-Godfrey test)
- Breusch-Godfrey (LM test = 3.51, df = 1, $\rho$ =0.48)

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q.S80)+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)
Model assumptions tested and verified (see Model Diagnostics)
- GVLMA (Global Stats = 1.75, $\rho$ =0.78)
Variance inflation factors (VIF) evaluated for model

Variable	VIF
QC44	2.35
QS308	2.15
ln(QS80)	3.53
Mean Lake Stage	2.88

Residuals check for residual autocorrelation (Breusch-Godfrey test)
- Breusch-Godfrey (LM test = 3.51, df = 1, $\rho$ =0.48)

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

S-80 total phosphorus model results and estimates using available data during the water year 1982 - 2019 period. Data were split into training and testing datasets (70:30).
	Estimate	Standard Error	t-value	ρ-value
(Intercept)	-2.49	0.68	-3.64	≤ 0.01
QC44	-2.85x10-7	7.07x10-7	-0.40	0.69
QS308	-5.29x10-8	2.35x10-7	-0.22	0.82
ln(QS80)	1.22	0.06	20.21	≤ 0.01
Mean Lake Stage	-0.13	0.05	-2.57	0.02

Residual standard error: 0.22 on 20 degrees of freedom
Multiple R-squared: 0.98, Adjusted R-squared: 0.98
F-statistic: 289.8 on 20 and 4, ρ-value: ≤ 0.01

ln(Q_S80) was included in the model to account for extreme variability in annual discharge and load. Exclusion of this parameter results in significant autocorrelation of the model residuals and loss of residual heteroscedasticity.

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

S-80 total phosphorus model results and estimates using available data during the water year 1982 - 2019 period. Data were split into training and testing datasets (70:30).
	Estimate	Standard Error	t-value	ρ-value
(Intercept)	-2.49	0.68	-3.64	≤ 0.01
QC44	-2.85x10-7	7.07x10-7	-0.40	0.69
QS308	-5.29x10-8	2.35x10-7	-0.22	0.82
ln(QS80)	1.22	0.06	20.21	≤ 0.01
Mean Lake Stage	-0.13	0.05	-2.57	0.02

Residual standard error: 0.22 on 20 degrees of freedom
Multiple R-squared: 0.98, Adjusted R-squared: 0.98
F-statistic: 289.8 on 20 and 4, ρ-value: ≤ 0.01

ln(Q_S80) was included in the model to account for extreme variability in annual discharge and load. Exclusion of this parameter results in significant autocorrelation of the model residuals and loss of residual heteroscedasticity.

$\begin{align*} ln(TP Load_{S80}) = -2.49 - (2.85x10^{-7}\times Q_{C44 Basin}) - (5.29x10^{-8}\times Q_{S308}) + (1.22\times ln(Q_{S80}))\\- (0.13\times Mean Stage) \end{align*}$

Model Diagnostics plots

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Relative importance of each predictor calculated by partitioning R² by averaging sequential sums of squares over all orders of regressors (Lindeman et al 1979). All metrics are normalized to a sum of 100%.

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Relative Importance Metrics for the S80 TP Load annual model.

Predictor	Percent of R²
QC44	13.5
QS308	14.8
ln(QS80)	56.2
Mean Lake Stage	15.6

Lindeman RH, Merenda PF, Gold RZ (1979) Introduction to bivariate and multivariate analysis. Scott Foresman & Co, Glenview, Illinois, USA

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TP loads at S-80 based on predictive model. Actual and predicted concentration were highly correlated (Spearman’s correlation: r=0.97, $\rho$ <0.01).

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TP loads at S-80 based on predictive model. Actual and predicted concentration were highly correlated (Spearman’s correlation: r=0.97, $\rho$ <0.01).

Model Fit

$R^{2}_{adj}$ : 0.98
RSE : 0.22

Train:Test

Mean Absolute Percent Error: 13 %
Min-Max Accuracy: 87 %
Nash-Sutcliffe Coefficient: 0.90
Kling-Gupta Coefficient: 0.86

Nash-Sutcliffe/Kling-Gupta explainations

Model RSE (backtransformed): 26739.11

Mean absolute percentage error - lower the better Min_Max Accuracy - higher the better Nash-Sutcliffe - 1 = perfect model (error variance divided by observed variance); https://en.wikipedia.org/wiki/Nash%E2%80%93Sutcliffe_model_efficiency_coefficient

Kling-Gupta - similar to NS range -1 to 1

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TP loads at S-80 with each k-model presented.

S-80 Water Quality Model (Total Phosphorus)

$\begin{align*} ln(TP Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TP loads at S-80 with each k-model presented.

k-fold (k=10)

Cross-validation error (average k errors)

	Parameter	Mean	Min	Max
Model	R2adj	0.97	0.96	0.98
Model	RMSE	0.20	0.17	0.23
Train:Test	MAPE 1	19	13	28
	MMA 1	84	78	88
	NS 2	0.93	0.89	0.97
	KG 2	0.86	0.74	0.98
1 Mean Absolute Percent Error (MAPE) and Min-Max Accuracy (MMA) expressed in percent
2 NS = Nash-Sutcliffe coefficient
2 KG = Kling-Gupta coefficient

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)
Model assumptions tested and verified (see Model Diagnostics)
- GVLMA (Global Stats = 6.67, $\rho$ =0.15)
Variance inflation factors (VIF) evaluated for model

Variable	VIF
QC44	2.35
QS308	2.15
ln(QS80)	3.53
Mean Lake Stage	2.88

Residuals check for residual autocorrelation (Breusch-Godfrey test)
- Breusch-Godfrey (LM test = 1.20, df = 1, $\rho$ =0.27)

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

TP load was log-transformed to fit the assumptions of linear modeling.
- Excluded WY2008 and WY2012 from analysis (No/low flow conditions)
Model assumptions tested and verified (see Model Diagnostics)
- GVLMA (Global Stats = 6.67, $\rho$ =0.15)
Variance inflation factors (VIF) evaluated for model

Variable	VIF
QC44	2.35
QS308	2.15
ln(QS80)	3.53
Mean Lake Stage	2.88

Residuals check for residual autocorrelation (Breusch-Godfrey test)
- Breusch-Godfrey (LM test = 1.20, df = 1, $\rho$ =0.27)

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

S-80 total nitrogen model results and estimates using available data during the water year 1982 - 2019 period. Data were split into training and testing datasets (70:30).
	Estimate	Standard Error	t-value	ρ-value
(Intercept)	1.76x10-2	0.51	0.03	0.97
QC44	6.60x10-8	5.24x10-7	0.13	0.90
QS308	1.99x10-7	1.74x10-7	1.14	0.27
ln(QS80)	1.06	0.04	23.66	≤ 0.01
Mean Lake Stage	-1.70x10-2	0.04	-0.47	0.65

Residual standard error: 0.16 on 20 degrees of freedom
Multiple R-squared: 0.99, Adjusted R-squared: 0.99
F-statistic: 510.9 on 20 and 4, ρ-value: ≤ 0.01

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

S-80 total nitrogen model results and estimates using available data during the water year 1982 - 2019 period. Data were split into training and testing datasets (70:30).
	Estimate	Standard Error	t-value	ρ-value
(Intercept)	1.76x10-2	0.51	0.03	0.97
QC44	6.60x10-8	5.24x10-7	0.13	0.90
QS308	1.99x10-7	1.74x10-7	1.14	0.27
ln(QS80)	1.06	0.04	23.66	≤ 0.01
Mean Lake Stage	-1.70x10-2	0.04	-0.47	0.65

Residual standard error: 0.16 on 20 degrees of freedom
Multiple R-squared: 0.99, Adjusted R-squared: 0.99
F-statistic: 510.9 on 20 and 4, ρ-value: ≤ 0.01

$\begin{align*} TN Load_{S79} = 1.76\times 10^{-2} + (6.60\times 10^{-8} Q_{C44 Basin}) + (1.99\times 10^{-7} Q_{S308}) + (1.06\times 10^{-2} ln(Q_{S80}))\\ - (1.70x10^{-2} Mean Stage) \end{align*}$

Model Diagnostics plots

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Relative Importance Metrics for the S80 TN Load annual model.

Predictor	Percent of R²
QC44	14.7
QS308	16.2
ln(QS80)	51.2
Mean Lake Stage	17.9

Lindeman RH, Merenda PF, Gold RZ (1979) Introduction to bivariate and multivariate analysis. Scott Foresman & Co, Glenview, Illinois, USA

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TN loads at S-80 based on predictive model. Actual and predicted concentration were highly correlated (Spearman’s correlation: r=0.96, $\rho$ <0.01).

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TN loads at S-80 based on predictive model. Actual and predicted concentration were highly correlated (Spearman’s correlation: r=0.96, $\rho$ <0.01).

Model Fit

$R^{2}_{adj}$ : 0.99
RSE : 0.16

Train:Test

Mean Absolute Percent Error: 22 %
Min-Max Accuracy: 83 %
Nash-Sutcliffe Coefficient: 0.93
Kling-Gupta Coefficient: 0.85

Model RSE (backtransformed): 102326

Mean absolute percentage error - lower the better Min_Max Accuracy - higher the better

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TN loads at S-80 with each k-model presented.

S-80 Water Quality Model (Total Nitrogen)

$\begin{align*} ln(TN Load_{S80}) = Q_{C44 Basin} + Q_{S308} + ln(Q_{S80})+ Mean Lake Stage \end{align*}$

Actual versus predicted TN loads at S-80 with each k-model presented.

k-fold (k=10)

Cross-validation error (average k errors)

	Parameter	Mean	Min	Max
Model	R2adj	0.97	0.96	0.99
Model	RMSE	0.19	0.14	0.24
Train:Test	MAPE 1	19	9	26
	MMA 1	85	80	92
	NS 2	0.93	0.88	0.99
	KG 2	0.88	0.74	0.96
1 Mean Absolute Percent Error (MAPE) and Min-Max Accuracy (MMA) expressed in percent
2 NS = Nash-Sutcliffe coefficient
2 KG = Kling-Gupta coefficient

S-80 Water Quality Models

Annual observed versus predicted ( $\pm$ 95% CI) S-80 load during the period of record (WY1982 – WY 2019) with hurricane years identified.

S-80 Water Quality Models

Annual observed versus predicted ( $\pm$ 95% CI) S-79 load during the period of record (WY1982 – WY 2019) with hurricane years identified.

Using monthly WQ data

Similar to CRE models, period of record monthly nutrient concentrations were considered
Other restoration planning efforts (i.e. Restoration Strategies) have used this method in the past.
Evaluated by comparing observed versus estimate (i.e. "predicted") by computing RMSE

Root Mean Square Error (RMSE)

$\begin{align*} RMSE = \sqrt{ \frac{\sum_{i=1}^{n}( X_{i} - \hat{X_{i}} )^{2} }{n} } \end{align*}$

$X_{i}$ : Oberseved value

$\hat{X_{i}}$ : Predicted value

$n$ : Number of observations

Month	Total Phosphorus (μg L⁻¹) a	Total Nitrogen (mg L⁻¹) a
Jan	106 ± 36 (64)	1.18 ± 0.32 (63)
Feb	108 ± 49 (66)	1.28 ± 0.41 (66)
Mar	114 ± 51 (67)	1.27 ± 0.47 (68)
Apr	117 ± 42 (66)	1.24 ± 0.47 (65)
May	132 ± 72 (67)	1.12 ± 0.34 (65)
Jun	178 ± 83 (59)	1.29 ± 0.4 (56)
Jul	216 ± 106 (63)	1.36 ± 0.45 (63)
Aug	197 ± 82 (69)	1.37 ± 0.67 (67)
Sep	221 ± 97 (65)	1.45 ± 0.38 (62)
Oct	192 ± 62 (62)	1.48 ± 0.41 (64)
Nov	155 ± 62 (69)	1.41 ± 0.45 (70)
Dec	113 ± 39 (62)	1.24 ± 0.29 (61)
a Mean ± Std Dev (N)
POR: Jan 1981 - April 2019
Station ID: C44S80
Data Source: SFWMD DBHydro

Monthly POR Estimates

Pair monthly mean TP and TN concentrations with discharge volumes to estimate load (See prior slide).

Monthly POR Estimates

How does POR estimates approach match up?

Root mean standard error for models and period of record estimates.
Model	Estimate Method	RMSE A B
TP Load	Model	23916
TP Load	POR Est.	36478
TN Load	Model	91523
TN Load	POR Est.	329708
A RMSE value for POR Est. calculated using observed values versus annual estimated values using monthly mean concentrations
B RMSE value for Model - All Data backcalculated on untransformed predicted and observed values

S-80 Water Quality Models

Comparison of observed, modelled and period of record estimated nutrient loads at S-80 between Florida Water Year 1982 - 2019 (May 1981 - April 2019).

S-80 Water Quality Models

Comparison of observed, modelled and period of record estimated nutrient flow-weighted mean at S-80 between Florida Water Year 1982 - 2019 (May 1981 - April 2019).

S-80 Water Quality Models

Application of model with RSM-BN outputs¹

¹Provisional RSM BN outputs with POR extension. For demonstration/testing purposes only.

S-80 Water Quality Models

Application of model with RSM-BN outputs¹

¹Provisional RSM BN outputs with POR extension. For demonstration/testing purposes only.

RSM Evalution

Compare loading conditions of selected alternatives to some base conditions (i.e. ECB, LORS08, etc).
Both models assume that C43 and C44 Reservoirs are providing temporary storage of existing/available water.
Both models do not incorporate potential water quality treatment features
- CRE: C43 Water Quality Feasibility Study project.
- SLE: C44 Reservoir and STA.
To evalute potential WQ improvements loading could be evaluated post processing in a Monte-Carlo like evaluation assuming a degree of treatment (i.e. % reduction, $X$ metric tons, etc.).

Acknowledgements

Data

South Florida Water Management District (DBHYDRO)

Slides

RMarkdown Source

Draft FDEP Work Product

DRAFT
28

S80 TP Model diagnostics

S80 TP model diagnostics plots (Top Left: Residuals vs Fitted, Bottom Left: Normal Q-Q, Top Right: Scale-Location, Bottom right: Residuals vs leverage.).

GVLMA (Global Stats = 1.75, $\rho$ =0.78)
Shapiro-Wilk normality test (W=0.94, $\rho$ =0.12)

S80 TP Model residual Autocorrelation Function.

Breusch-Godfrey (LM test = 1.58, df = 1, $\rho$ =0.21)

TP Model plots

S80 TN Model diagnostics

S80 TN model diagnostics plots (Top Left: Residuals vs Fitted, Bottom Left: Normal Q-Q, Top Right: Scale-Location, Bottom right: Residuals vs leverage.).

GVLMA (Global Stats = 6.68, $\rho$ =0.27)
Shapiro-Wilk normality test (W=0.98, $\rho$ =0.92)

S80 TN Model residual Autocorrelation Function.

Breusch-Godfrey (LM test = 1.20, df = 1, $\rho$ =0.27)

TN Model plots

Model Efficiency Coefficient

Nash-Sutcliffe

$\begin{align*} NS = 1- \frac{\sum_{t=1}^{n} (X_{s,t} - X_{o,t})^2}{\sum_{t=1}^{n} (X_{o,t} - \mu_{o})^2} \end{align*}$

$n$ : total number of time-steps
$X_{s,t}$ : simulated value at timestep $t$
$X_{o,t}$ : observed value at timestep $t$
$\mu_{o}$ : mean of observed values

The ratio of error variance of the modeled versus observed timeseries

Kling-Gupta

$\begin{align*} KG = 1- \sqrt{(r_{pearson}-1)^{2} + \left(\frac{\sigma_{s}}{\sigma_{o}}-1 \right)^{2} + \left(\frac{\mu_{s}}{\mu_{o}}-1 \right)^2} \end{align*}$

$r_{pearson}$ : Pearson correlation coefficient
$\mu_{s}$ : mean of simulated values
$\sigma_{o}$ : standard deviation of observed values
$\sigma_{s}$ : standard deviation of simulated values

Decomposition of NS representing the degree of correlation, bias and variablity of simulated and observed values.

↑, ←, Pg Up, k	Go to previous slide
↓, →, Pg Dn, Space, j	Go to next slide
Home	Go to first slide
End	Go to last slide
Number + Return	Go to specific slide
b / m / f	Toggle blackout / mirrored / fullscreen mode
c	Clone slideshow
p	Toggle presenter mode
t	Restart the presentation timer
?, h	Toggle this help